A firm uses two inputs, clean air services and stuff, to make an output, electricity. Clean air services is the use of the air to take away the firms products of combustion, like carbon monoxide. It is a garbage disposal service. Stuff is just shorthand for coal, labor, boilers, and the like. The price of clean air services without government intervention is taken to be very low (but not zero, since it costs something to throw the waste products into the air.) Government has a choice of how to regulate the firm. It can change the price of clean air services or it can mandate a certain method of production and quantity of output.
The firms technology is summarized in its isoquants. Below is a diagram showing how a cost-minimizing firm would choose the amounts of clean air and stuff to use in making q* units of output. For simplicity set w2=1. Since all points on an isocost line cost the same and the vertical intercept is a point on a line, the cost of the vertical intercept (a point of the form (0,x2)) is the cost associated with the isocost line. If an isocost line hits the vertical axis at x2* then the associated cost is w2 x2*= x2*. The diagram shows two isocost lines, the steeper one has w1 higher than the less steep one. They are labelled with their prices w1 or x1, clean air. Notice that the one with the higher price of air uses less air to make q* than the one with the lower price of air.
Assuming that the government knew the correct price for clean air (9.2/8 in the diagram), then 2.5 is the correct number of units of clean air to use in making q*. Since the government (in this example) knows the firms isoquants, couldn't the government just specify that 2.5 units of clean air were to be used when the output would be q*? The answer is yes, if the government knew the output, the regulation could be use only 2.5 units. The difference between the price (clean air will cost you 9.2/8 per unit) and quantity regulation (use 2.5 units) is that the firm has lower costs with the quantity regulation. Under both regulatory regimes the firm chooses input mix (2.5,6.3). Under the price regime (government induces the input choice by setting w1=9.2/8) the cost is 9.2. Under the quantity regime(the government forces the use of 2.5 units of air) the cost is 2.5*w1 + 6.3 or 2.5*11/51 + 6.3 = 6.8. Note that this calculation is carried out with the unregulated cost of air, w1=10.4/51. Draw a line parallel to the low cost line (w1=10.4/51) and through (2.5,6.3) and you should find that its vertical intercept is 6.8, which is a graphical way of finding the cost of the quantity regulated output. The conclusion:Quantity regulation costs the firm less than the price regulation, at any given choice of q and the same level of pollution.
In the simple case where the tax and the quantity regulation lead to the same chosen input bundle, the marginal cost is the same at that point (and only at that point.). Lets call that output q*. So far, I only know of a calculus proof of this assertion. For the case where the factor demand for effluent is increasing in output, you can draw two isoquants with lower output than q* and show that MC under regulation is less than MC under tax. For the case where it is more than q*, the result is the opposite. Putting this together, by continuity the two marginal costs must be equal at q*.
Does Industry Output go Up?
If firms are allowed to enter the industry and new firms receive the same effluent standards as the older firms, then the long run supply curve will be flat. Since the costs are lower under effluent standards, the long run supply curve will be lower, price will be lower and industry output and pollution greater!